Remainder Regions

Remainder Regions gives you a map split into regions, each showing a number, and a small palette of colours. Pick a colour, tap a region to fill it, and tap again with the same colour to clear it. Fill every region and you have a candidate answer.

One rule decides whether it is correct, and it depends on the numbers. Two regions that touch must be given different colours when the difference between their numbers is prime. Regions showing 24 and 29 differ by 5, and 5 is prime, so those two must not match. Regions showing 24 and 24 differ by 0, which is not prime, so they are free — they may share a colour or not, whichever helps elsewhere. That asymmetry is the whole puzzle: a non-prime difference is permission, never an obligation, so a colour you place to satisfy one border is still yours to choose for the next.

Depending on the level the rule can also be about remainders — regions whose numbers leave the same remainder when divided by 3, say — or plain map colouring with no arithmetic at all. Every board is solvable with the colours provided. It suits students who have met prime numbers and are ready to use them for something other than a factor tree.